Self-duality in the class of precompact groups

Abstract

A topological Abelian group G is called ( strongly) self-dual if there exists a topological isomorphism Φ : G → G ∧ of G onto the dual group G ∧ (such that Φ ( x ) ( y ) = Φ ( y ) ( x ) for all x , y ∈ G ). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κ ω = κ , there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.